This field of study is also called just dynamical systems, mathematical dynamical systems theory or the mathematical theory of dynamical systems.
Dynamical systems theory and chaos theory
deal with the long-term qualitative behavior of dynamical systems
. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible steady states?", or "Does the long-term behavior of the system depend on its initial condition?"
An important goal is to describe the fixed points, or steady states of a given dynamical system; these are values of the variable that don't change over time. Some of these fixed points are attractive, meaning that if the system starts out in a nearby state, it converges towards the fixed point.
Similarly, one is interested in periodic points
, states of the system that repeat after several timesteps. Periodic points can also be attractive. Sharkovskii's theorem
is an interesting statement about the number of periodic points of a one-dimensional discrete dynamical system.
Even simple nonlinear dynamical systems
often exhibit seemingly random behavior that has been called chaos
The branch of dynamical systems that deals with the clean definition and investigation of chaos is called chaos theory
The concept of dynamical systems theory has its origins in Newtonian mechanics
. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is given implicitly by a relation that gives the state of the system only a short time into the future.
Before the advent of fast computing machines
, solving a dynamical system required sophisticated mathematical techniques and could only be accomplished for a small class of dynamical systems.
A dynamical system has a state
determined by a collection of real numbers
, or more generally by a set
in an appropriate state space
. Small changes in the state of the system correspond to small changes in the numbers. The numbers are also the coordinates of a geometrical space—a manifold
. The evolution rule
of the dynamical system is a fixed rule
that describes what future states follow from the current state. The rule may be deterministic
(for a given time interval one future state can be precisely predicted given the current state) or stochastic
(the evolution of the state can only be predicted with a certain probability).
, a nonlinear system
is a system that is not linear
—i.e., a system that does not satisfy the superposition principle
. Less technically, a nonlinear system is any problem where the variable(s) to solve for cannot be written as a linear sum of independent components. A nonhomogeneous
system, which is linear apart from the presence of a function of the independent variables
, is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed to a linear system as long as a particular solution is known.
describes the behavior of certain dynamical systems
– that is, systems whose state evolves with time – that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect
). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears random
. This happens even though these systems are deterministic
, meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved. This behavior is known as deterministic chaos, or simply chaos
is a scientific field that studies the common properties of systems
, and science
. It is also called complex systems theory
, complexity science
, study of complex systems
and/or sciences of complexity
. The key problems of such systems are difficulties with their formal modeling
. From such perspective, in different research contexts complex systems are defined on the base of their different attributes.
The study of complex systems is bringing new vitality to many areas of science where a more typical reductionist
strategy has fallen short. Complex systems
is therefore often used as a broad term encompassing a research approach to problems in many diverse disciplines including neurosciences
, social sciences
, computer science
, artificial life
, evolutionary computation
, earthquake prediction, molecular biology
and inquiries into the nature of living cells
Graph dynamical systems
The concept of graph dynamical systems
(GDS) can be used to capture a wide range of processes taking place on graphs or networks. A major theme in the mathematical and computational analysis of graph dynamical systems is to relate their structural properties (e.g. the network connectivity) and the global dynamics that result.
Projected dynamical systems
is an approach to understanding the behaviour of systems over time. It deals with internal feedback loops and time delays that affect the behaviour and state of the entire system.
What makes using system dynamics different from other approaches to studying systems is the use of feedback
loops and stocks and flows
. These elements help describe how even seemingly simple systems display baffling nonlinearity
is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of general topology
In sports biomechanics
, dynamical systems theory has emerged in the movement sciences as a viable framework for modeling athletic performance and efficiency. From a dynamical systems perspective, the human movement system is a highly intricate network of co-dependent sub-systems (e.g. respiratory, circulatory, nervous, skeletomuscular, perceptual) that are composed of a large number of interacting components (e.g. blood cells, oxygen molecules, muscle tissue, metabolic enzymes, connective tissue and bone). In dynamical systems theory, movement patterns emerge through generic processes of self-organization found in physical and biological systems.
There is no research validation of any of the claims associated to the conceptual application of this framework.
In cognitive science
Dynamical system theory has been applied in the field of neuroscience
and cognitive development
, especially in the neo-Piagetian theories of cognitive development
. It is the belief that cognitive development is best represented by physical theories rather than theories based on syntax and AI
. It also believed that differential equations are the most appropriate tool for modeling human behavior. These equations are interpreted to represent an agent's cognitive trajectory through state space
. In other words, dynamicists argue that psychology
should be (or is) the description (via differential equations) of the cognitions and behaviors of an agent under certain environmental and internal pressures. The language of chaos theory is also frequently adopted.
In it, the learner's mind reaches a state of disequilibrium where old patterns have broken down. This is the phase transition of cognitive development. Self-organization
(the spontaneous creation of coherent forms) sets in as activity levels link to each other. Newly formed macroscopic and microscopic structures support each other, speeding up the process. These links form the structure of a new state of order in the mind through a process called scalloping
(the repeated building up and collapsing of complex performance.) This new, novel state is progressive, discrete, idiosyncratic and unpredictable.
Dynamic systems theory has recently been used to explain a long-unanswered problem in child development referred to as the A-not-B error
In second language development
The application of Dynamic Systems Theory to study second language acquisition
is attributed to Diane Larsen-Freeman
who published an article in 1997 in which she claimed that second language acquisition
should be viewed as a developmental process which includes language attrition
as well as language acquisition.
In her article she claimed that language should be viewed as a dynamic system which is dynamic, complex, nonlinear, chaotic, unpredictable, sensitive to initial conditions, open, self-organizing, feedback sensitive, and adaptive.
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Last edited on 12 May 2021, at 02:28
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